Question

Упростить тригонометрические уравнения.

Answer 1

Решение:

1.

$\dfrac{sin^2\alpha }{1 - sin^2\alpha} \cdot ctg^2\alpha = \dfrac{sin^2\alpha }{cos^2\alpha} \cdot \dfrac{cos^2\alpha}{sin^2\alpha} = 1$

2.

$2sin~t + \dfrac{2cos^2t}{1 + sin~t} = \dfrac{2sin~t + 2sin^2t + 2 cos^2t}{1 + sin~t}=\dfrac{2sin~t + 2}{1 + sin~t } = \dfrac{2(1 + sin~t)}{1 + sin~t} =2$

3.

$2sin^2~2\alpha + cos~4\alpha = 2 sin^2~2\alpha + cos^2~2 \alpha - sin^2~2\alpha = sin^2~2\alpha + cos^2~2 \alpha = 1$

4.

$(sin~\alpha + cos~\alpha)^2 - sin~2\alpha = sin^2~\alpha + 2 sin~\alpha \cdot \cos~\alpha + cos^2~\alpha - sin~2\alpha =\\=1 + sin~2\alpha - sin~2\alpha = 1$

5.

$\dfrac{2sin^2\alpha}{1-cos~2\alpha} = \dfrac{2sin^2\alpha}{sin^2\alpha + cos^2\alpha-cos^2\alpha+sin^2\alpha} = \dfrac{2sin^2\alpha}{2sin^2\alpha} = 1$